Asim Patra

National Institute of Technology Rourkela | NITR · Department of Mathematics (MA)

This paper displays the approach of the time-splitting Fourier spectral (TSFS) technique for the linear Riesz fractional Schrödinger equation (RFSE) in the semi-classical regime. The splitting technique is shown to be unconditionally stable. Further a suitable implicit finite difference discretization of second order has been manifested for the RFS...

A balancing-like sequence is a binary recurrence sequence which generalizes the balancing sequence and the sequence of nonnegative integers. This sequence, under certain assumptions, may be used to describe the growth of fortune of a person engaged in some business or profession. Since, in any business or profession, the growth is influenced by man...

In this study, the approximation of the higher order-linear Fredholm integro-differential-difference equations (IDDEs), with the mixed conditions, have been performed by a new collocation technique based on the balancing polynomials. In particular, an attempt has been made to transform the linear IDDEs and the given boundary conditions into matrix...

In this study we have tackled with some nonlinear identities and Diophantine equations related to balancing-like and Lucas-balancing-like numbers. Apart from that, the equation x^4 - 8C_n x^2 y + 16 y^2 = -2^r has been considered for the investigation of existence of any positive integer solutions x and y which is the leftover case dealt in Patra a...

In this work, a reliable and efficient numerical technique viz. the balancing collocation technique (BCT) has been introduced and employed to solve the linear two-dimensional Fredholm-Volterra integral (F-VI) equations. The technique reduces the solution of these integral equations to the solution of a linear system of algebraic equations. Furtherm...

In this work, we study sums of finite products of Pell polynomials and express them in terms of some special orthogonal polynomials. Furthermore, each of the obtained expression is represented as linear combinations of classical polynomials involving hypergeometric functions by means of explicit computations.

We study certain subsequences of the balancing numbers and determine their residues modulo the powers of the balancing numbers. This leads to an exact divisibility theorem for the subsequence and also gives an analogous result to those of the Fibonacci numbers.

In this work, the Catalan transformation (CT) of k-balancing sequences, 􏽮Bk,n􏽯n≥0, is introduced. Furthermore, the obtained Catalan transformation CB􏽮 k,n􏽯n≥0 was shown as the product of lower triangular matrices called Catalan matrices and the matrix of k-balancing sequences, B􏽮 k,n􏽯n≥0, which is an n × 1 matrix. Apart from that, the Hankel transf...

The balancing and Lucas-balancing numbers are solutions of second order recurrence relations. A linear combination of these numbers can also be obtained as solutions of a fourth order recurrence relation. This recurrence relation can be extended to generalized quaternion algebras. Also, the fourth order recurrence relation has application in coding...

Let \(P_n\) be the n-th Pell number and \(Q_n\) be the n-th Associated Pell number. We obtain certain divisibility and exact divisibility results for the powers of the Pell and Associated Pell numbers by applying the concept of p-adic valuation of the numbers. In particular, we show that \(P_n^k \parallel m\) if and only if \(P_n^{k+1} \parallel P_...

The present article deals with the similarity method to tackle the fractional Schrӧdinger equation where the derivative is defined in the Riesz sense. Moreover, the procedure of reducing a fractional partial differential equation (FPDE) into an ordinary differential equation (ODE) has been efficiently displayed by means of suitable scaled transform...

The balancing numbers x and the Lucas-balancing numbers y are solutions of the Diophantine equation 8x^2 + 1 = y^2 , and both types of numbers satisfy a common second order recurrence relation. These numbers can be seen as numerators and denominators in the steady state probabilities of a class of Markov chains.

This paper is devoted to solving the equations x s − 8Cnxy + 16y t = ±2 r for (s, t) ∈ {(2, 2), (2, 4), (4, 2)} in positive integers x and y. The solutions are obtained in terms of the balancing, Pell and Lucas-Pell numbers.